How would I go about finding the distribution function from this. I looked at the bernoulli distribution, I didnīt really see how I could use it for my particular problem though. Help on this area would be appreciated.

Also, my attempt on the expected value gave me .

Could this be correct? It seems reasonable that the expected value tends to zero as , but also that the expected value is not zero when t is small, because it is more likely that is even for small t.

And this is exactly a Bernoulli distribution over {-1,1} with parameter p(t)
Why do you absolutely want a distribution function ? You can write it if you really want to, but it's not really necessary...